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Consider the path fibration $PB \to B$, assuming you've chosen a base-point in $B$. This map is null homotopic. A null map $PB \to B$ lifts to unique base-point preserving map $PB \to E$ (constant the base-point in $F$). So provided your fibration satisfies the homotopy lifting property for the special case of the p... |
Consider the path fibration $PB \to B$, assuming you've chosen a base-point in $B$. This map is null homotopic. A constant map $PB \to B$ lifts to unique base-point preserving map $PB \to E$ (constant the base-point in $F$). So provided your fibration satisfies the homotopy lifting property for the special case of t... |
Im familiar with the notation $Ord_a(x)$, when $x$ is an integer. However, Im reading a book where this notation is used with $x$ rational. Im not sure how to interpret this? |
Most of the answers so far have been along the general lines of 'Why hard problems are important', rather than 'Why the Collatz conjecture is important'; I will try to address the latter.
I think the basic question being touched on is:
> In what ways does the prime factorization of $a$ affect the prime factorizat... |
Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with periodic function... |
Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with periodic function... |
Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the origin and the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with pe... |
I figured that I can create a deterministic "random" numbers generator by utilizing a bit of "magic" that I picked up from some cryptography. However I seem to have missed a detail.
Basically the function `f(n) = p^n mod q` should for `n` in the range `[1;q-1]` produce all the numbers in the same range, but in a scr... |
Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the origin and the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with pe... |
How can I prove that if G is an abelian group with elements a and b with orders m and n, respectively, then G contains an element whose order is the least common multiple of m and n?
It's an exercise from Hungerford's book, but it's not homework. I could not solve it when I take the course on groups and I think it s... |
Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the origin and the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with pe... |
I figured that I can create a deterministic "random" numbers generator by utilizing a bit of "magic" that I picked up from some cryptography. However I seem to have missed a detail.
Basically the function $\displaystyle f(n) = p^n \mod q$ should for $\displaystyle n$ in the range $\displaystyle [1,q-1]$ produce all ... |
Given an open set $U \subset \mathbb R ^n $, there exists an exhaustion by compact sets, i.e. a sequence of compact sets $K_i$, s.t.
$\cup _{i=0}^{\infty} K_i = U$ and $\forall i \in \mathbb N : K_i \subset K_{i+1} ^{\circ}$
We can imagine that different exhaustions by compact sets 'propagate' at different speeds... |
I'm working on a program that displays a bunch of thumbnails on the bottom of the screen and I'm using the mouse's position to slide the images from side to side.
The center of the screen is 0 on an x coordinate plane. Every frame, the mouse's X position is either added or subtracted from the x position of each thu... |
I'm working on a program that displays a bunch of thumbnails on the bottom of the screen and I'm using the mouse's position to slide the images from side to side.
The center of the screen is 0 on a number line. Every frame, the mouse's X position is either added or subtracted from the x position of each thumbnail. ... |
I'm working on a program that displays a bunch of thumbnails on the bottom of the screen and I'm using the mouse's position to slide the images from side to side.
The center of the screen is 0 on a number line. Every frame, the mouse's X position is either added or subtracted from the x position of each thumbnail. ... |
Functions from $\mathbb{C} \rightarrow \mathbb{C}$ are hard to visualize because of their 4-dimensional nature. One nice way of looking at them is by what's called [domain coloring][1]. An example from the wiki article is shown below.
When we look at the graph of a real function ($\mathbb{R} \rightarrow \mathbb{R}$)... |
What can be gleaned from looking at a domain-colored graph of a complex function? |
Below is an inductive proof excerpted from my [post here.][1]
**LEMMA** $\ $ A finite abelian group $\rm\:G\:$ has an lcm-closed order set, i.e. with $\rm\: o(X) = $ order of $\rm\: X$
$\rm\quad\quad\quad\quad\quad\quad\ X,Y \in G\ \ \Rightarrow\ \ \exists\ Z \in G:\ o(Z) = lcm(o(X),o(Y))$
**Proof** $\ \ $ ... |
If we have a module M over a ring Z, and we consider the the canonical module homomorphism M→Q⊗M over Z, is it true that the kernel of this map is the torsion submodule of M? Why? |
I'm familiar with the notation $\mathrm{ord}_a(x)$, when $x$ is an integer. However, I'm reading a book where this notation is used with $x$ rational. I'm not sure how to interpret this? |
In the proof of Stone-Weierstrass theorem provided in Rudin's Principles of Mathematical analysis, why do we only need to show that there exists a sequence of polynomials Pn that converges uniformly to the continuous complex function on [0 1]? |
In the proof of Stone-Weierstrass theorem provided in Rudin's Principles of Mathematical Analysis, why do we only need to show that there exists a sequence of polynomials $P_n$ that converges uniformly to the continuous complex function on $[0,1]$? |
So in algebraic geometry, the prime spectrum of the ring of polynomials over a field k is isomorphic to the set of k-homomorphisms from the ring of polynomials to k, and we do this by using the fact that the residue field (localise w.r.t. some prime ideal, then mod out the ideal generated by this in the localisation) i... |
A space is regular if each closed set $Z$ is the intersection of all open sets containing $Z$? |
I was reading Rudin's Principles of Mathematical Analysis, and I came across the definition 7.19, where it says that a sequence of functions $f_n(x)$ is pointwise bounded on E if there exists a finite-valued function $\phi$ defined on E such that
$$ |f_n(x)| < \phi(x) $$
for x element of E, n = 1, 2 ,3 ...
While $... |
Why doesn't pointwise bounded imply uniform bounded? |
When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number? |
A manifold M is simply connected if for every pair of 1-cubes c1,c2: [0,1] -> M with
c1(0) = c1(1) = c2(0) = c2(1) = t
there is a 2-cube b such that
1) b1,0 = c1 and b1,1 = c2
2) for all p in [0,1] b2,0(p) = b2,1(p) = t
My question is, if I'm given a manifold M,
for example choose M = S^2 = {(x,y,z)... |
I am given two random variables Y1 and Y2 and:
E(Y1) = 4
E(Y2) = -1
V(Y1) = 2
V(Y2) = 8
I am asked to find Cov(Y1,Y1)
I know Cov(Y1,Y2) = E(Y1Y2)-E(Y1)E(Y2)
I'm not sure if it is a typo, but it says Cov(Y1,Y1) NOT Cov(Y1,Y2)
The answer is 2, but i'm not sure how to get E(Y1Y2) from the information given... |
When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number?
I'm looking for an answer deeper than "there are 6*60=360 degrees in a circle, so you can picture it". |
**HINT:** When do you say that a set is bounded.
- *A set $X \subset \mathbb{R}$ is said to be bounded if you can find a $M \in \mathbb{R}$ such that $|x| \leq M$ for all $x \in X$.*
Now consider your cases:
- $X = \{ x | x = \frac{1}{n} ; \ n \in \mathbb{N}\}$ **NOTE:** $\frac{1}{n} \leq 1$ for all $n \in ... |
Isn't the trivial (indiscrete) topology a counterexample? Perhaps there are more assumptions somewhere. |
Suppose that there is a certain collected works of plays that is N symbols long in the following sense: a "symbol" is one of the 26 letters of the alphabet, a line breaks, period, semicolon, or a colon; in other words there are 30 possible symbols.
If "a monkey" randomly "types" 1 of these 30 symbols at a rate of on... |
Suppose that there is a certain collected works of plays that is N symbols long in the following sense: a "symbol" is one of the 26 letters of the alphabet, a line breaks, period, space, or a colon; in other words there are 30 possible symbols.
If "a monkey" randomly "types" 1 of these 30 symbols at a rate of one pe... |
Suppose that there is a certain collected works of plays that is N symbols long in the following sense: a "symbol" is one of the 26 letters of the alphabet, a line break, period, space, or a colon; in other words there are 30 possible symbols.
If "a monkey" randomly "types" 1 of these 30 symbols at a rate of one per... |
Isn't the trivial (indiscrete) topology a counterexample? Perhaps there are more assumptions somewhere.
Edit: I had a different definition of "regular" in mind from the asker. The question has been clarified, and the trivial topology is not a counterexample. |
Which of the following is the unbounded set ?
1. $ X = \{ x\mid x = \frac{1}{n},n \in \mathbb{N} \} $
2. $ Y = \{ x\mid x = \frac{1}{2^n},n \in \mathbb{N} \} $
3. $ Z = \{ x\mid x = 2^n,n \in \mathbb{N} \} $
4. $ W = \{ x\mid x \in \mathbb{N}, x \lt 4532 \} $
After reading the definition from [... |
How to solve this manually ?
|
Suppose that there is a certain collected works of plays that is N symbols long in the following sense: a "symbol" is one of the 26 letters of the alphabet, a line break, period, space, or a colon; in other words there are 30 possible symbols.
If "a monkey" randomly "types" 1 of these 30 symbols at a rate of one per... |
How to find the value of
$$\displaystyle\sqrt{3} \cdot \cot (20^{\circ}) - 4 \cdot \cos (20^{\circ})$$
manually ?
|
In a $\displaystyle\bigtriangleup$ ABC,R is circumradius and $\displaystyle 8R^2 = a^2 + b^2 + c^2 $ , then $\displaystyle\bigtriangleup$ ABC is of which [type][2] ?
[2]: http://en.wikipedia.org/wiki/Triangle#By_internal_angles |
What type of triangle satisfies: $8R^2 = a^2 + b^2 + c^2 $ ? |
This is from the book Principles of Mathematical Analysis by Rudin, number 4 of chapter 7. It says consider
$$ f(x) = \sum\limits_{n=1}^{\infty}{ 1/(1+ n^2 x) } $$
The question asks:
(1) For what values of x does the series converge absolutely. We got that the series converges when x $\not=$ 0 & x $\not= -1/... |
This is from the book Principles of Mathematical Analysis by Rudin, number 4 of chapter 7. It says consider
$$ f(x) = \sum\limits_{n=1}^{\infty}{ 1/(1+ n^2 x) } $$
The question asks:
(1) For what values of x does the series converge absolutely. We got that the series converges when x $\not=$ 0 & x $\not= -1/... |
I have a question about a transformation of a matrix. Lets say we have the following matrix
$
M =
\left[ {\begin{array}{cc}
4 & 3 \\
4 & 3 \\
\end{array} } \right]
$
Then I want to transform all numbers in the matrix with the following function:
$f(x) = (x - 4)^2 + 2x -4$
What would be the corre... |
**HINT:**
You have seen a number of discontinuities at certain points.
1. Can you use the M-test between points of discontinuities?
1.a If you can then it is good.
1.b If you can not do that - why? Can you do estimates if you remove a part around these points?
2. Can you show that the convergence is NOT uni... |
I realized that your question wasn't exactly about the Steenrod axioms themselves, but about the definition of the coboundary operator involving suspension. In reduced homology, the boundary operator $\partial$ for the pair $(X,A)$ (where the inclusion $i:A\rightarrow X$ is a cofibration) can be defined to come from t... |
Maybe I don't understand your question, but: what's wrong with this?
$$
\begin{pmatrix}
f(4) & f(3) \\\
f(4) & f(3)
\end{pmatrix}
$$
More generally, if you have a matrix $(a^i_j)$, you could write the resulting matrix as $(f(a^i_j))$. |
At Wikipedia I found that "[according to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs][1]".
> Are there infinitely many asymmetric
> *planar* cubic graphs, too?
> If so, does it follow that there is an infinite asymmetric planar cubic graph?
**Background**
I... |
At Wikipedia I found that "[according to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs][1]".
> Are there infinitely many asymmetric
> *planar* cubic graphs, too?
> If so, does it follow that there is an *infinite* asymmetric planar cubic graph?
**Background**
... |
Reposting from stackoverflow :) - Told to go here :)
So I've got a issue with a math assignment, and I were hoping someone could at least point me to what I should be doing, because currently I'm just wasting my time, thats for sure. - Please don't solve it all, thats up for me to do so :)
The assignment sounds l... |
At Wikipedia I found that "[according to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs][1]".
> Are there infinitely many asymmetric
> *planar* cubic graphs, too?
> If so, does it follow that there is an *infinite* asymmetric planar cubic graph?
> If so, how coul... |
$G$ is a finite abelian group. $A \times B$ can be embedded in $G$.
Does this mean there exist $C$, $D$ such that $G=C \times D$ and $A$ can be embedded in $C$ and $B$ can be embedded in $D$? |
$G$ finite abelian. $A \times B$ embedded in $G$. Is $G=C \times D$ such that $A$ embedded in $C$, $B$ embedded in $D$? |
I figured that I can create a deterministic "random" numbers generator by utilizing a bit of "magic" that I picked up from some cryptography. However I seem to have missed a detail.
Basically the function $\displaystyle f(n) = p^n \mod q$ should for $\displaystyle n$ in the range $\displaystyle [1,q-1]$ produce all ... |
Define a subprobability measure to be a measure on Borel sigma algebra of the real line $\mathbb{R}$ with the measure for the whole real line less or equal than 1.
I was wondering about the definition of vague convergence of a sequence of subprobability measures $\{ \mu_n, n\geq 1 \}$ to another subprobablity measu... |
Define a subprobability measure to be a measure on Borel sigma algebra of the real line $\mathbb{R}$ with the measure for the whole real line less or equal than 1.
I was wondering about the definition of vague convergence of a sequence of subprobability measures $\{ \mu_n, n\geq 1 \}$ to another subprobablity measu... |
Define a subprobability measure to be a measure on Borel sigma algebra of the real line $\mathbb{R}$ with the measure for the whole real line less or equal than 1.
I was wondering about the definition of vague convergence of a sequence of subprobability measures $\{ \mu_n, n\geq 1 \}$ to another subprobablity measu... |
Is it possible to define distribution function for a measure on a general measurable space, which generalizes the definition of distribution function for a probability measure on the extended real line?
If yes, how?
If no, how general/special can the measure and the measurable space be in order to define its di... |
YARFMO (Yet another reposting from Mathoverflow) ;-)
The more you know about math the more you find conceptions previously thought correct to be false:
1.) math is not as exact as many believe - in many cases, as Gowers points out in his amazing book "Mathematics. A very short intro", it is mainly about approxima... |
YARFMO (Yet another reposting from Mathoverflow) ;-)
The more you know about math the more you find conceptions previously thought correct to be false:
1.) math is not as exact as many believe - in many cases, as Gowers points out in his amazing book "Mathematics. A very short intro", it is mainly about approxima... |
YARFMO (Yet another reposting from Mathoverflow) ;-)
The more you know about math the more you find conceptions previously thought correct to be false:
1.) math is not as exact as many believe - in many cases, as Gowers points out in his amazing book "Mathematics. A very short intro", it is mainly about approxima... |
I was wondering what is wrong in the following proof:
Proposition. Let $\{f_{n}\}$ be a sequence of integrable functions such that $f_{n}$ converges pointwise to a function $f$. Show that if:
$lim \int |f_{n} - f| d\mu =0$ then $\int |f| d\mu = \lim \int |f_{n}| d\mu$.
Well I used the fact that $||f_{n}| - |... |
Let $f: A \rightarrow B$ be a ring homomorphism between commutative rings with identity. Then there exists an induced map $f' : Spec(B) \rightarrow Spec(A)$. If $f'$ is surjective, then clearly every prime ideal of $A$ is a contracted ideal. Now my question is, is the converse true? |
If every prime ideal is a contracted ideal, does it imply that the induced map between spectrums is surjective? |
I figured that I can create a deterministic "random" numbers generator by utilizing a bit of "magic" that I picked up from some cryptography. However I seem to have missed a detail.
Basically the function $\displaystyle f(n) = p^n \mod q$ should for $\displaystyle n$ in the range $\displaystyle [1,q-1]$ produce all ... |
Given $1<q\leq2$ and $0\leq p\leq1$, let us consider the following function:
$\phi\left(\alpha\right)=p\times\left|1-\alpha\right|^{q}+\left(1-p\right)\times\left|1+\alpha\right|^{q}$
The minimum over $\mathbb{R}$ I have found is:
$\alpha\left(p\right)=\frac{p^{\frac{1}{q-1}}-\left(1-p\right)^{\frac{1}{q-1}}}{p^{\... |
I guess this might as well be a separate answer. You can use quadratic reciprocity to give elementary proofs of certain special cases of <a href="http://en.wikipedia.org/wiki/Dirichlets_theorem_on_arithmetic_progressions">Dirichlet's theorem</a>. First, you should be aware of the following nice result and its "Euclid... |
I guess this might as well be a separate answer. You can use quadratic reciprocity to give elementary proofs of certain special cases of <a href="http://en.wikipedia.org/wiki/Dirichlets_theorem_on_arithmetic_progressions">Dirichlet's theorem</a>. First, you should be aware of the following nice result and its "Euclid... |
This is a problem in baby Rudin. Chapter 4, problem 17. I'm just looking at examples at the moment. It says that any continuous open map is monotone. Recall, an open map f: X -> Y is such that any open set U in X will haves its image, f(U) open in Y.
So,
Let $$ f: \mathbb{R} \to [-1, 1] $$ take $$ x \mapsto sin(x) ... |
How do I show that a monomorphism $F \rightarrow G$ of sheaves induces an injection on stalks?
When showing that monomorphism is an injection on sets one uses the maps $x \mapsto a$ and $x \mapsto b$ where $a,b$ are some fix elements such that they have the same image under the monomorphism. Then by using the monom... |
I was wondering what is wrong in the following proof:
Proposition. Let $\{f_{n}\}$ be a sequence of integrable functions such that $f_{n}$ converges pointwise to a function $f$. Show that if:
$\lim \int |f_{n} - f| d\mu =0$ then $\int |f| d\mu = \lim \int |f_{n}| d\mu$.
Well I used the fact that $||f_{n}| - |f... |
This is a problem in baby Rudin. Chapter 4, problem 15. I'm just looking at examples at the moment. It says that any continuous open map is monotone. Recall, an open map f: X -> Y is such that any open set U in X will haves its image, f(U) open in Y.
So,
Let $$ f: \mathbb{R} \to [-1, 1] $$ take $$ x \mapsto sin(x) ... |
If $\alpha$ is large (of either sign), $\phi$ increases without bound essentially as $\alpha ^q$. Is $\alpha \in (-1,1)$ in which case we can delete the absolute value bars? |
Perhaps more meta than you wanted since it's not about how you perceive mathematics coming from mathematics, but rather how the rest of the world views it.
There's that whole misconception that if you're good at math you're good at arithmetic. You know... that thing where someone says something like "Hey, you're goo... |
Given $1<q\leq2$ and $0\leq p\leq1$, let us consider the following function:
$\phi\left(\alpha\right)=p\times\left|1-\alpha\right|^{q}+\left(1-p\right)\times\left|1+\alpha\right|^{q}$
The minimum over $\mathbb{R}$ I have found is:
$\alpha\left(p\right)=\frac{p^{\frac{1}{q-1}}-\left(1-p\right)^{\frac{1}{q-1}}}{p^{\... |
Here are a couple of misconceptions that people who don't know about mathematics commonly hold. Firstly, that it's immutable, fixed, unchanging; surely it's all been discovered? It's a belief that even a highly educated non-mathematician might hold.
This point was illustrated to me not long ago when a friend, who sp... |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
Let $K \subset R^n$ be compact, and let $C_0^{\infty}(K)$ be the space of smooth functions on with support in $K$.
For $p \in [1,\infty), \alpha$ multiindex, let $|f|_{\alpha,p} = (\int_K |D^{\alpha}f|^p)^{\frac{1}{p}}$, and $|f|_{\alpha,\infty} = sup_K |D^{\alpha}f|$ - as usual.
For $p \in [1,\infty]$, let $\tau... |
The centres of three circles of radius r just touching one another will form an equilateral triangle of side length 2r. As we keep adding another circle of radius r around the edge of the central one, another such triangle will be formed. Exactly 6, non-overlapping equilateral triangles with be formed in this way as yo... |
The centres of three circles of radius r just touching one another will form an equilateral triangle of side length 2r. Exactly 6 such non-overlapping equilateral triangles will be formed in this way as you keep adding more circles around the edge of the central circle. Why 6? These equilateral triangles will have a co... |
I've noticed it's quite common among algebraists dealing with finite structures (like finite groups) to think of mathematics (limited to their area, of course) as the absolute God given truth, harmonious, non-contradictory, sort of Platonic ideal world.
This point of view can be possible accepted, as long we limit o... |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
How to sketch a sinc function by hand? |
http://www.wolframalpha.com/input/?i=plot+sinc%28x%29 |
I think its helpful also to graph it with $\frac{1}{x}$
[http://www.wolframalpha.com/input/?i=plot+sinc%28x%29%2C+1%2Fx][1]
[1]: http://www.wolframalpha.com/input/?i=plot+sinc%28x%29%2C+1%2Fx
If you write the function as $f(x) = \frac{\sin(x)}{x}$ for $x \not=0$ and $1$ if $x=0$ you can see that it will b... |
In answering a question I mentioned the Asteroids video game as the example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It might be out of date in 2010. This raises its own question:
are there other real-life examples of geometries formed... |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
After asking a few questions earlier, I think I've been able to describe my main issue. Could someone show me why this identity is true:
$$\frac{\partial \textbf{x}^{T} B \textbf{x}}{\partial \textbf{x}} = (B + B^{T})\textbf{x}$$
I've gone through things like Matrixcookbook which seem to just show this identity ... |
Problem: Calculate limit of (1/sqrt(n+1) + 1/sqrt(n+2) + 1/sqrt(n+3) + ... + 1/sqrt(n+n^2)) as n approaches infinite.
Solution: Denote the above some as X, then we can bound it:
infinite <---- 1/sqrt(n+n^2) < X < n^2/sqrt(n+1) < n^2/sqrt(n) = sqrt(n^4/n) ----> infinite
So, from the Squeeze Principle, lim x =... |
In answering a question I mentioned the Asteroids video game as the example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It might be out of date in 2010. This raises its own question:
are there other real-life examples of geometries formed... |
Problem: Calculate limit of $\frac{1}{\sqrt{n+1}} + \frac{1}{\sqrt{n+2}} + \frac{1}{\sqrt{n+3}} + \cdots + \frac{1}{\sqrt{n+n^2}}$ as $n$ approaches infinity.
Solution: Denote the above some as $X$, then we can bound it:
$$ \infty\longleftarrow\frac{n^2}{\sqrt{n+n^2}} \lt X \lt \frac{n^2}{\sqrt{n+1}} \lt \frac{n^2}... |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
Problem: Calculate limit of $\frac{1}{\sqrt{n+1}} + \frac{1}{\sqrt{n+2}} + \frac{1}{\sqrt{n+3}} + \cdots + \frac{1}{\sqrt{n+n^2}}$ as $n$ approaches infinity.
Solution: Denote the above some as $X$, then we can bound it:
$$ \infty\longleftarrow\frac{1}{\sqrt{n+n^2}} \lt X \lt \frac{n^2}{\sqrt{n+1}} \lt \frac{n^2}{\... |
Let $\mu (r)$ be the [irrationality measure](http://mathworld.wolfram.com/IrrationalityMeasure.html) of a real number $r$, and consider the following set of points $P \in\mathbb{R}$:
$P=\{r\in \mathbb{R}: \mu(r)=Constant\}$
Is this set a fractal, and If so, then what is it's dimension?
|
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